On the positive side, our work unveils the potential statistical improvements offered by non-linear predictors. In the theorem statement above, \(\widehat{f}\)  denotes a modified Vovk-Azoury-Warmuth forecaster [1]}, [2]} introduced in [3]}. In addition to the \(\sqrt{d}\)  gap between (REF ) and (REF ), the non-linear predictor \(\widehat{f}\)  surpasses the lower bound (REF ) that holds for any algorithm returning a linear predictor in \(\mathcal {W}_{b}\) . In particular, it removes the dependence on the boundedness constants \(b\)  and \(r\)  appearing in the lower bound (REF ) completely removing any dependence on the distribution of the covariates.
